Digging around the Internet, I learned that Android Jellybean has a new security feature. You need to do RSA key fingerprint verification with the PC. Otherwise the Android device will be reported as offline.

I had the issue that aufter 6h of using adb (was working well) the device turned offline.If you do not allow the device it is not called offeline it says "unauthorized"There are 3 Status messages messages "unauthorized" "device" and "offline"What you describe here is how to solve the "unauthorized" problem. but not the "offline" problem.